





'''simple docstring'''






from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray






def   __A    (				a_    :	float   ,a_    :	float   ,a_    :	bool = False						):
		if radian_mode:
				return [magnitude * cos(a_						), magnitude * sin(a_						)]
		return [magnitude * cos(radians(a_						)						), magnitude * sin(radians(a_						)						)]






def   __A    (				a_    :	NDArray[floataa]   ,a_    :	NDArray[floataa]   ,a_    :	float = 1_0**-1						):
		lowerCAmelCase  :     NDArray[floataa]       						= cross(a_   ,a_						)
		lowerCAmelCase  :     float       						= sum(a_						)
		return abs(a_						) < eps


if __name__ == "__main__":
					# Test to check if it works
					lowerCAmelCase									=   array(
					    [
					        polar_force(718.4, 1_80 - 30),
					        polar_force(879.54, 45),
					        polar_force(1_00, -90),
					    ]
					)

					lowerCAmelCase									=   array([[0, 0], [0, 0], [0, 0]])

					assert in_static_equilibrium(forces, location)

					# Problem 1 in image_data/2D_problems.jpg
					lowerCAmelCase									=   array(
					    [
					        polar_force(30 * 9.81, 15),
					        polar_force(2_15, 1_80 - 45),
					        polar_force(2_64, 90 - 30),
					    ]
					)

					lowerCAmelCase									=   array([[0, 0], [0, 0], [0, 0]])

					assert in_static_equilibrium(forces, location)

					# Problem in image_data/2D_problems_1.jpg
					lowerCAmelCase									=   array([[0, -20_00], [0, -12_00], [0, 1_56_00], [0, -1_24_00]])

					lowerCAmelCase									=   array([[0, 0], [6, 0], [10, 0], [12, 0]])

					assert in_static_equilibrium(forces, location)

					import doctest

					doctest.testmod()










'''simple docstring'''






from __future__ import annotations

lowerCAmelCase									=   []






def   __A    (				a_    :	list[list[int]]   ,a_    :	int   ,a_    :	int						):
		for i in range(len(a_						)						):
				if board[row][i] == 1:
						return False
		for i in range(len(a_						)						):
				if board[i][column] == 1:
						return False
		for i, j in zip(range(a_   ,-1   ,-1						)   ,range(a_   ,-1   ,-1						)						):
				if board[i][j] == 1:
						return False
		for i, j in zip(range(a_   ,-1   ,-1						)   ,range(a_   ,len(a_						)						)						):
				if board[i][j] == 1:
						return False
		return True






def   __A    (				a_    :	list[list[int]]   ,a_    :	int						):
		if row >= len(a_						):
				solution.append(a_						)
				printboard(a_						)
				print()
				return True
		for i in range(len(a_						)						):
				if is_safe(a_   ,a_   ,a_						):
						lowerCAmelCase  :     Dict       						= 1
						solve(a_   ,row + 1						)
						lowerCAmelCase  :     Optional[int]       						= 0
		return False






def   __A    (				a_    :	list[list[int]]						):
		for i in range(len(a_						)						):
				for j in range(len(a_						)						):
						if board[i][j] == 1:
								print("Q"   ,end=" "						)
						else:
								print("."   ,end=" "						)
				print()


# n=int(input("The no. of queens"))
lowerCAmelCase									=   8
lowerCAmelCase									=   [[0 for i in range(n)] for j in range(n)]
solve(board, 0)

print("""The total no. of solutions are :""", len(solution))



